166 THEORY OF HEAT. [CHAP. III. 



SECTION V. 



Finite expression of the result of the solution^ 



205. The preceding solution might be deduced from the 



d 2 v d*v 

 integral of the equation -y~ 2 + -3-3 = O, 1 which contains imaginary 



quantities, under the sign of the arbitrary functions. We shall 

 confine ourselves here to the remark that the integral 



v=&amp;lt;!&amp;gt;(x+yj -T) +^r(x- W^T), 

 has a manifest relation to the value of v given by the equation 



-T- = e~ x cos y ^ e~ Zx cos 3y -f ^ e~ 5x cos oy &c. 

 4 o 5 



In fact, replacing the cosines by their imaginary expressions, 

 we have 



- &c. 

 3 o 



The first series is a function of x yJ\, and the second 

 series is the same function of x + yj 1. 



Comparing these series with the known development of arc tan z 

 in functions of z its tangent, it is immediately seen that the first 

 is arc tan e if ** f3r \ and the second is arc tan e ^^ ; thus 

 equation (a) takes the finite form 



~ = arc tan e - (x+v ^ + arc tan e -&amp;lt;*- v=r &amp;gt; 



In this mode it conforms to the general integral 



v = &amp;lt;t&amp;gt;(x + yj~\) + ^(x-yj~^l) ......... (A), 



the function $ (z) is arc tan e~&quot;, and similarly the function i|r (z). 

 1 D. F. Gregory derived the solution from the form 



Cumb. Math. Journal, Vol. I. p. 105. [A. F.] 



